Natural Number Concepts Research Articles

Mathematical Concepts Exploration of Gamelan Musical Instruments and Dance Movements in Kuda Lumping Wanareja

This research explored mathematical concepts of Gamelan musical instruments and dance movements in Kuda Lumping Wanareja. This research used a qualitative approach with an ethnographic design. Located of research at Cihandiwung Lor, Adimulya, Wanareja, Cilacap, Central Java. Data was collected through observation, interviews, and documentation. Data were analyzed through reduction, presentation, and verification of conclusions. The research results showed that Wanareja's Kuda Lumping gamelan musical instruments consisted of kendang, demung, saron, calapita, kenong, bonang, kethuk, drum, gong, and kempul. Based on the shape of gamelan musical instruments, mathematical concepts include numbers, measurement, and geometry. The saron, demung, bonang, and kempul contain the concept of natural numbers, while kempul shows the concept of quantity. The kendang, demung, saron, calapita, kenong, bonang, kethuk, drum, gong, and kempul show the concept of a rectangle, circle, trapezoid, and tube. The layout of gamelan musical instruments (especially bonang) involves a matrix concept. The Kuda Lumping dance consists of ngatur sembah, pecak pedhang, losing to the enemy, fighting against the enemy from the right, fighting against the enemy from the left, ending the battle, mejang beksa, and janturan/mendem. The mathematical concept related to dance movement patterns is geometric transformation (reflection, translation, and rotation). Several dance movements for praying, breaking the sword, losing to the enemy, fighting against the enemy, and ending the battle show reflection and translation. The janturan and mejang beksa dance movements show rotation. Concerning Gamelan musical instruments, there are kethuk and saron which show two-dimensional geometric concepts such as circles and trapezoids. The results of this research are applied as context in mathematical problems.

The Picture Book and its Role in Preschool Mathematics Education

The paper examines the role of the picture book as a learning resource in preschool mathematics education and illustrates the possibilities for its application. The paper aims to examine whether preschool teachers use picture books in preschool mathematics education, what areas of mathematics they most use these for, and whether they recognize methodological advantages to their application. Results show that half of preschool teachers use the picture book as a teaching resource in preschool mathematics education, that it is mostly used to develop the concept of natural numbers and spatial relations, as well as that teachers clearly perceive methodological advantages.

Arithmetic thinking as the basis of children's generative number concepts

Predominant psychological theories of number acquisition posit that children acquire natural number concepts as they acquire the successor principle, or the knowledge that every natural number is succeeded by another natural number that is exactly-one more than it. However, exactly how children acquire the successor principle remains largely unexplained. Recently developed ideas within this family of theories posit that an abstract recursive successor function is acquired from the recursive structure of number words; however, the types of recursion underlying the successor function and number words are distinctively different (one is a self-referential function and the other is a self-embedded structure), making it difficult to theorize how one type triggers the acquisition of another. Moreover, our analysis of the literature questions if the knowledge about the successor principle is even empirically measurable. Here, we argue that number acquisition is a process of understanding a generative rule that governs the system of natural numbers and point out that the successor principle is not the only generative rule that governs the natural number system. We propose an alternative hypothesis that generative number concepts emerge from children's realization about how the combinatorial rules of numerals allow arithmetic (specifically additive and multiplicative) representations of quantity. Importantly, under addition and multiplication—which are historically rooted in concatenation and grouping of physical objects—natural numbers are mathematically closed. As a corollary, the system of infinitely generative natural numbers is conceptualized. This new theoretical framework allows the construction of novel empirical questions and testable hypotheses based on the formalized rules of numerical syntax and numeration systems, and therefore opens a new avenue for studying later stages of children's acquisition of number concepts.

ETNOMATHEMATICS: EXPLORATION OF THE ART OF KENTHONGAN MUSIC AS A SOURCE OF MATHEMATICS LEARNING

Ethnomathematics is an approach tio learning mathematics that connects the material studied with the local culture of students. Ethnomathematics can be used by mathematics educators to overcome problems in the field of mathematics, namely the negative perception of students about mathematics which is seen as difficult. This study aims to explore, identify, describe and analyze mathematical concepts contained in the art of kenthongan music. This research is a qualitative descriptive exploratory research with an ethnographic approach. Data were collected by observation, documentation, and interviews. This research was conducted on the kenthongan music art group "Syailendra". The analytical technique used is the Miles and Huberman model and the data validity test uses triangulation. This study shows that there are mathematical concepts found in the art of kenthongan music, especially in the kenthongan art group "Syailendra" which can be used as learning resources in learning mathematics. The mathematical concepts found include the concept of geometry in which there is the concept of a circle and the concept of a tube, then there is the concept of natural numbers and the concept of arithmetic sequences. Furthermore, universal mathematical activities such as counting, locating, measuring, designing, playing, and explaining activities were also found

On Number-Set Identity: A Study

Abstract Benacerraf’s 1965 multiple-reductions argument depends on what I call ‘deferential logicism’: his necessary condition for number-set identity is most plausible against a background Quineanism that allows autonomy of the natural number concept. (This explains his neglect of ‘theoretical-virtue’ arguments.) Steinhart’s ‘folkist’ sufficient condition on number-set identity, by contrast, puts that autonomy at the center — but fails for not taking the folk perspective seriously enough. Learning from both sides, we explore new conditions on number-set identity, elaborating a 1983 suggestion from Wright.

Epistemic and Mediational Suitability of Tasks Designed in a Mobile Learning Context

The aim of this research is to propose indicators of epistemic and mediational suitability to assess tasks designed in a mobile learning context, using theoretical and methodological elements of the onto-semiotic approach (OSA) and the SAMR model (substitution, augmentation, modification, and redefinition). With these indicators, 110 tasks designed by two Mexican teachers are evaluated, for the teaching of the natural number in primary education. Some results indicate that the situation-problems addressed the meanings of numerical sequence, operational, ordinal, and measure, and that 56% of the tasks were located at the substitution level. Some implications derived from the research are that the mathematical content can be complemented by including other representations of the natural number and that the design of tasks that are positioned at the modification and redefinition levels should be encouraged. The foregoing serves as a basis for the design and assessment of mobile content for the natural number concept.

Predicting children's emerging understanding of numbers.

How do children construct a concept of natural numbers? Past research addressing this question has mainly focused on understanding how children come to acquire the cardinality principle. However, at that point children already understand the first number words and have a rudimentary natural number concept in place. The question therefore remains; what gets children's number learning off the ground? We therefore, based on previous empirical and theoretical work, tested which factors predict the first stages of children's natural number understanding. We assessed if children's expressive vocabulary, visuospatial working memory, and ANS (Approximate number system) acuity at 18 months of age could predict their natural number knowledge at 2.5 years of age. We found that early expressive vocabulary and visuospatial working memory were important for later number knowledge. The results of the current study add to a growing body of literature showing the importance of language in children's learning about numbers.

The emergence of children’s natural number concepts: Current theoretical challenges

AbstractLearning the meaning of number words is a lengthy and error‐prone process. In this review, we highlight outstanding issues related to current accounts of children’s acquisition of symbolic number knowledge. We maintain that, despite the ability to identify and label small numerical quantities, children do not understand initially that number words refer only to sets of discrete countable items, not to other nonnumerical dimensions. We question the presence of a sudden change in children’s understanding of cardinality, and we report the limits of the give‐a‐number task. We also highlight that children are still learning the directional property of the counting list, even after acquiring the cardinality principle. Finally, we discuss the role that the Approximate Number System may have in supporting the acquisition of symbolic numbers. We call for improvements in methodological tools and refinement in theoretical understanding of how children learn natural numbers.

Cognitive Structuralism: Explaining the Regularity of the Natural Numbers Progression

According to one of the most powerful paradigms explaining the meaning of the concept of natural number, natural numbers get a large part of their conceptual content from core cognitive abilities. Carey’s bootstrapping provides a model of the role of core cognition in the creation of mature mathematical concepts. In this paper, I conduct conceptual analyses of various theories within this paradigm, concluding that the theories based on the ability to subitize (i.e., to assess an exact quantity of the elements in a collection without counting them), or on the ability to approximate quantities (i.e., to assess an approximate quantity of the elements in a collection without counting them), or both, fail to provide a conceptual basis for bootstrapping the concept of an exact natural number. In particular, I argue that none of the existing theories explains one of the key characteristics of the natural number structure: the equidistances between successive elements of the natural numbers progression. I suggest that this regularity could be based on another innate cognitive ability, namely sensitivity to the regularity of rhythm. In the final section, I propose a new position within the core cognition paradigm, inspired by structuralist positions in philosophy of mathematics.

Implicit and Explicit Examples of the Phenomenon of Deviant Encodings

Abstract The core of the problem discussed in this paper is the following: the Church-Turing Thesis states that Turing Machines formally explicate the intuitive concept of computability. The description of Turing Machines requires description of the notation used for the input and for the output. Providing a general definition of notations acceptable in the process of computations causes problems. This is because a notation, or an encoding suitable for a computation, has to be computable. Yet, using the concept of computation, in a definition of a notation, which will be further used in a definition of the concept of computation yields an obvious vicious circle. The circularity of this definition causes trouble in distinguishing on the theoretical level, what is an acceptable notation from what is not an acceptable notation, or as it is usually referred to in the literature, “deviant encodings”. Deviant encodings appear explicitly in discussions about what is an adequate or correct conceptual analysis of the concept of computation. In this paper, I focus on philosophical examples where the phenomenon appears implicitly, in a “disguised” version. In particular, I present its use in the analysis of the concept of natural number. I also point at additional phenomena related to deviant encodings: conceptual fixed points and apparent “computability” of uncomputable functions. In parallel, I develop the idea that Carnapian explications provide a much more adequate framework for understanding the concept of computation, than the classical philosophical analysis.

The Interplay Between the Natural Number Bias and Fraction Magnitude Processing in Low-Achieving Students

Research has identified two core difficulties many students have with fractions: First, they often struggle with processing fraction magnitudes, and second, they rely on natural number concepts in fraction problems (“Natural Number Bias”, NNB). Yet, the relation between these two difficulties is not well understood. Moreover, while most studies of the NNB relied on analyses of the whole samples, there is empirical evidence that the occurrence of the NNB differs between student subgroups. In the present study, we investigate individual students’ profiles of the occurrence of the natural number bias and their ability to process fraction magnitude, using a dynamic assessment that utilizes continuous diagrams on touchscreen devices. We analyze data of 234 low-achieving 6th-grade students from Germany who completed a symbolic fraction comparison task, and a fraction magnitude estimation task with continuous circle and tape diagrams. A cluster analysis on the comparison task revealed three distinct clusters: a Typical Bias cluster (better performance on symbolic fraction comparison items congruent to natural number-based reasoning), a Reverse Bias cluster (better performance on items incongruent to natural number-based reasoning), and a No Bias cluster (similar performance on congruent and incongruent items). Only students in the No Bias cluster but not students in the other clusters demonstrated a distance effect in symbolic fraction comparison, suggesting fraction magnitude processing. Linear mixed models on the percent absolute error in the magnitude estimation task revealed significantly lower percent absolute error for students in the No Bias cluster compared to students in the other two clusters. Moreover, students in the No Bias cluster were significantly slower to solve both fraction comparison and fraction magnitude estimation tasks than students in the other clusters. These results suggest that the occurrence of the natural number bias and the ability to process fraction magnitude are closely related. The continuous representations used in our digital assessment tools appeared to be suitable for assessing both the natural number bias and fraction magnitude processing.

Natural Alternatives to Natural Number: The Case of Ratio.

The overwhelming majority of efforts to cultivate early mathematical thinking rely primarily on counting and associated natural number concepts. Unfortunately, natural numbers and discretized thinking do not align well with a large swath of the mathematical concepts we wish for children to learn. This misalignment presents an important impediment to teaching and learning. We suggest that one way to circumvent these pitfalls is to leverage students' non-numerical experiences that can provide intuitive access to foundational mathematical concepts. Specifically, we advocate for explicitly leveraging a) students' perceptually based intuitions about quantity and b) students' reasoning about change and variation, and we address the affordances offered by this approach. We argue that it can support ways of thinking that may at times align better with to-be-learned mathematical ideas, and thus may serve as a productive alternative for particular mathematical concepts when compared to number. We illustrate this argument using the domain of ratio, and we do so from the distinct disciplinary lenses we employ respectively as a cognitive psychologist and as a mathematics education researcher. Finally, we discuss the potential for productive synthesis given the substantial differences in our preferred methods and general epistemologies.

AN EXAMPLE OF PREPARED-PLANNED CREATIVE DRAMA IN SECOND GRADE MATHEMATICS EDUCATION

The aim of this research is teaching addition with natural numbers and the concept of large and small natural numbers in the second grade mathematics course, through creative drama method. The study has been applied to 31 elementary school second grade students studying at a public school in the province of Aydin. In this research, case study based on observation method, which is one of the qualitative research methods, has been employed. The data have been collected through drama lesson plans, observation, and interview methods. The activities involving the acquisitions of ''adding two-digit natural numbers'' and ''absorbing large-small concepts'' prepared by the researcher, have been applied in the classroom. The drawings of students, literary notes, and observation reports kept by researchers have been used in the evaluation of the activity. As a result of this application, students' views towards the questions covering the whole application have been gathered. The results of the study suggest that prepared-planned creative drama activities are an entertaining and enjoyable way for students to learn. The notes taken by the participative observers and findings obtained from the interviews indicate that drama activities increase sharing among students and that the students could associate what they have learned with some of the situations they encounter in their every day-life. It is observed that the students learn by having more fun while playing games, which involves a high amount of physical movements. Drawing the attention of students through instructions in the warm-up phase of the activity increased students' curiosity in learning. In the animation phase of the activity, it was observed that the students' interest in the mathematics increased, they had fun while making the mathematical operations, their participation to the lesson was high throughout the activity, the energy they received from the lesson was at the maximum level, they learned by doing and living, and they obtained an experience they could use in their every day-life. In the evaluation phase of the activity, the students were asked to describe how they felt by drawing a picture or writing a story as a summary of the creative drama activity they experiences by doing. According to the stories they have written and the pictures they have drawn, it is apparent that they were entertained and happy throughout the activity. The results of this study indicate that the students relieve their mental exhaustion and a relaxed classroom setting is achieved through the relaxation activities. Article visualizations:

Core Knowledge, Language, and Number

ABSTRACTThe natural numbers may be our simplest, most useful, and best-studied abstract concepts, but their origins are debated. I consider this debate in the context of the proposal, by Gallistel and Gelman, that natural number system is a product of cognitive evolution and the proposal, by Carey, that it is a product of human cultural history. I offer a third proposal that builds on aspects of these views but rejects one tenet that they share: the thesis that counting is central to number. I suggest that children discover the natural numbers when they learn a natural language: especially nouns, number words, and the rules that compose quantified noun phrases. This learning, in turn, depends both on cognitive systems that are innate and shared by other animals, and on our species-specific language faculty. Thus, natural number concepts are unique to humans and culturally universal, yet they are learned.

Mastery of the logic of natural numbers is not the result of mastery of counting: evidence from late counters.

To master the natural number system, children must understand both the concepts that number words capture and the counting procedure by which they are applied. These two types of knowledge develop in childhood, but their connection is poorly understood. Here we explore the relationship between the mastery of counting and the mastery of exact numerical equality (one central aspect of natural number) in the Tsimane', a farming-foraging group whose children master counting at a delayed age and with higher variability than do children in industrialized societies. By taking advantage of this variation, we can better understand how counting and exact equality relate to each other, while controlling for age and education. We find that the Tsimane' come to understand exact equality at later and variable ages. This understanding correlates with their mastery of number words and counting, controlling for age and education. However, some children who have mastered counting lack an understanding of exact equality, and some children who have not mastered counting have achieved this understanding. These results suggest that understanding of counting and of natural number concepts are at least partially distinct achievements, and that both draw on inputs and resources whose distribution and availability differ across cultures.

Preschool spontaneous focusing on numerosity predicts rational number conceptual knowledge 6 years later

Recent evidence suggests that early natural number knowledge is a predictor of later rational number conceptual knowledge, even though students’ difficulties with rational numbers have also been explained by the overuse of natural number concepts—often referred to as the natural number bias. Hannula and Lehtinen (Learn Instr 15:237–256, 2005) have shown that children’s tendency to spontaneously focus on numerosity (SFON) predicts the development of natural number and arithmetic skills. The present study follows 36 children from the age of 6 years to the age of 12 years in order to determine how preschool SFON tendency and number sequence skills are related to rational number conceptual knowledge at the age of 12 years. The results show that children’s SFON tendency before school age is a strong predictor of later rational numbers conceptual knowledge, even after controlling for preschool number sequence skills. This finding has implications for the understanding of how the transition from reasoning about natural number concepts to reasoning about rational numbers may be influenced by children’s self-initiated practice with numbers in everyday situations.

Toward exact number: Young children use one-to-one correspondence to measure set identity but not numerical equality

Exact integer concepts are fundamental to a wide array of human activities, but their origins are obscure. Some have proposed that children are endowed with a system of natural number concepts, whereas others have argued that children construct these concepts by mastering verbal counting or other numeric symbols. This debate remains unresolved, because it is difficult to test children’s mastery of the logic of integer concepts without using symbols to enumerate large sets, and the symbols themselves could be a source of difficulty for children. Here, we introduce a new method, focusing on large quantities and avoiding the use of words or other symbols for numbers, to study children’s understanding of an essential property underlying integer concepts: the relation of exact numerical equality. Children aged 32–36months, who possessed no symbols for exact numbers beyond 4, were given one-to-one correspondence cues to help them track a set of puppets, and their enumeration of the set was assessed by a non-verbal manual search task. Children used one-to-one correspondence relations to reconstruct exact quantities in sets of 5 or 6 objects, as long as the elements forming the sets remained the same individuals. In contrast, they failed to track exact quantities when one element was added, removed, or substituted for another. These results suggest an alternative to both nativist and symbol-based constructivist theories of the development of natural number concepts: Before learning symbols for exact numbers, children have a partial understanding of the properties of exact numbers.

Mental Magnitudes and Increments of Mental Magnitudes

There is at present a lively debate in cognitive psychology concerning the origin of natural number concepts. At the center of this debate is the system of mental magnitudes, an innately given cognitive mechanism that represents cardinality and that performs a variety of arithmetical operations. Most participants in the debate argue that this system cannot be the sole source of natural number concepts, because they take it to represent cardinality approximately while natural number concepts are precise. In this paper, I argue that the claim that mental magnitudes represent cardinality approximately overlooks the distinction between a magnitude and the increments that compose to form that magnitude. While magnitudes do indeed represent cardinality approximately, they are composed of a precise number of increments. I argue further that learning the number words and the counting routine may allow one to mark in memory the number of increments that composed to form a magnitude, thereby creating a precise representation of cardinality.

Das competências quantitativas iniciais para o conceito de número natural: quais as trilhas possíveis?

The present review aims to analyze the proposal put forward by researches on mathematical cognition about the initial quantitative abilities in human infants. Also, it analyses how the human infants may construct the concept of natural number upon these initial abilities. From the data presented, the paper discusses whether the idea of number has an innate basis, which is an original nativist's position, or a developmental process. Additionally, it presents data from new studies that point towards a cognitive process in the natural number concept that goes from general to specific domains. However, more studies are needed to support this proposal.

The prehistory of number concept

AbstractCarey leaves unaddressed an important evolutionary puzzle: In the absence of a numeral list, how could a concept of natural number ever have arisen in the first place? Here we suggest that the initial development of natural number must have bootstrapped on a material culture scaffold of some sort, and illustrate how this might have occurred using strings of beads.